I don't know if you'd consider this a "natural" explanation, but...
Kabbalistic sources speak of ten basic modes through which G-d relates to us, called sefiros. These are described in terms of three intellectual and seven emotional attributes (and indeed, they are the spiritual sources of these thoughts and emotions as we experience them). The seven "emotional" sefiros, then, in their various combinations and permutations, make up the complete gamut of how we experience G-d's presence in our world (and also how we relate to each other).
Further, these same sources describe four progressive spiritual "worlds," in each of which G-dliness is more and more concealed; the last of these is our own world, where G-d's presence is barely recognizable - the universe appears to function on its own without a Creator at all, G-d forbid - and it takes considerable thought to see the truth behind the natural order. Each of these four "worlds" has its own version of the ten sefiros, again progressively coarsened until they assume the forms with which we're familiar.
Thus, to echo Yaakov Ellis's point, the underlying reason why 7 represents a natural cycle is because in terms of the sefiros, 7 is a complete group of them (the three "intellectual" attributes find their expression through them, just as in a human being, one's intellectual ideas have to be mediated to others through one's emotional qualities). So, for example, each day of the week reflects a different "emotional" mode in which we relate to G-d and He to us; similarly, each day of the mourning period is meant for a person to process their grief through the prism of that day's sefirah attribute; and so forth.
Forty is, again as Yaakov noted, a number representing complete transformation. This is arrived at by taking the total number of sefiros in all four stages of their spiritual evolution through the Four Worlds - in other words, the total number of (describable) ways in which we perceive G-dliness. (Here both the "intellectual" and the "emotional" sefiros come into play, because they are both co-equal parts of the human psyche and of its spiritual source/analogue.) Thus, for example, to receive the Divine Torah, Moses had to spend forty days transforming all of these forty aspects of himself and his relationship with G-d, literally becoming a new man in the process. So too, an impure person immerses in a pool of 40 se'ah of water (and the earth, during the Flood, had to undergo that number of days of cleansing with water), and comes out as a renewed being.